#6814740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,25,39)(16,20,28,36)(17,30,31,21)(19,32,34,26)(24,38,27,42)(29,44,40,37)(33,43,35,41);(2,3)(5,7)(6,9)(8,12)(10,11);(1,2,4,3)(13,15,17,26,18,25,31,32)(14,19,21,23,22,34,30,39)(16,27,29,43,28,24,40,41)(20,35,37,42,36,33,44,38);(6,10)(9,11)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,37)(22,36)(23,38)(25,27)(26,41)(30,44)(31,40)(32,43)(34,35)(39,42);(1,3,4,2)(7,12)(9,11)(15,25)(16,28)(19,34)(20,36)(23,39)(26,32)(29,40)(37,44)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721936);